Here you go!
1880 U.S. Census, Oak Forest TWP, Lee County, Arkansas population schedule; Enumeration District 167, lines 20-26, Roll: T9_49. AC 5
Moore, J.H., 46, farmer, b. NC, father b. NC, mother b. NC
Susan, 36, wife, keeping house, NC, NC, NC
Elaine, 12, daughter, at school, AR, NC, NC
Norma, 10, daughter, at school, AR, NC, NC
Emit, 6, son, at school, AR, NC, NC
Lillian, 8 (or 5), daughter, at school, AR, NC, NC
Georgie, 1, daughter, AR, NC, NC
This soldier certainly paid his dues (Camp Douglas).
As with any genealogical pursuit, we often have to grab at straws to find information one can either prove or disprove later. One of your straws might be the 1850 Pitt County, NC, census, in which a John H. Moore appears with at least three known matching criteria (name, age and state of nativity).
1850 U.S. Census, Perkins TWP, Pitt County, North Carolina population schedule, Stamped Page 65, lines 20-28, Roll: M432_641;
James S. Moore, 44, farmer, b. NC (James Spencer Moore)
Isabella, 28, NC (Isabella Barnhill Moore)
Guilford, 10, NC
McGilbra, 8, NC
Charlotte T., 6, NC
Lawrence Cherry, 4, NC
Emilie, 2, NC
John H., 17, NC, Laborer
Lydia, 90, NC
Unfortunately, I was unable to locate John in the 1860 census, but that is not a rare thing. He might have been in transit. Equally unfortunate is that I did not find James S. Moore in the 1860 NC census to verify whether John H. was still in the household. I did find James in the 1880 NC census.
I did not find any Arkansas land patent records for John.
The only way to verify a connection is to visit repositories in the hope you can find documents that connect James and John (will).
If you wish to investigate local records on John, the courthouse in Helena should have something from between 1862 and 1872 (marriage record, tax records, court records). From 1873 onward, his records would be in Marianna, Lee County (possible will, tax records. court records).
Please bear in mind that the only thing definitely connected to John H. Moore in this message is the 1880 Lee County Census. Everything else is a straw that might open some doors.
Good luck in your search.